Verify Points on curve secp256k1

Question

I am trying to verify whether or not these points are on the secp256k1 curve.

I am finding several points included below.
(I have verified 2*G, 8*G and 10*G with the pycoin script)

My Questions are:

1. Did the sum of 2*G and inverse of 10*G come out correctly?
   If so, I am confused, because I thought that each X-Coordinate only had 2 Y solutions. (+Y) and (-Y)
   
   
2. When I plug all of these points into the curve equation, they all appear to lie on the curve. If the Summation Point is not on the curve, can someone please walk me through how to properly figure out whether or not a point is on the curve?
   
   

(Hex, then Decimal format)
2*G
(x) : 0C6047F9441ED7D6D3045406E95C07CD85C778E4B8CEF3CA7ABAC09B95C709EE5
(y) : 1AE168FEA63DC339A3C58419466CEAEEF7F632653266D0E1236431A950CFE52A
(x) : 89565891926547004231252920425935692360644145829622209833684329913297188986597
(y) : 12158399299693830322967808612713398636155367887041628176798871954788371653930

10*G(inverse)
(x): 0A0434D9E47F3C86235477C7B1AE6AE5D3442D49B1943C2B752A68E2A47E247C7
(y): F76C545BDABE643D85C4938196C5DB3969086B3D127885EA6C3411AC4FC8C9729
(x): 72488970228380509287422715226575535698893157273063074627791787432852706183111
(y): -62070622898698443831883535403436258712770888294397026493185421712108624767191

Point Summation - 2*G + 10*G(Inverse)
(x): 2F01E5E15CCA351DAFF3843FB70F3C2F0A1BDD05E5AF888A67784EF3E10A2A01
(y): 0A3B25758BEAC66B6D6C2F7D5ECD2EC4B3D1DEC2945A489E84A25D3479342132B
(x): 21262057306151627953595685090280431278183829487175876377991189246716355947009
(y): 74042095941090708372193120377072390691273078602492804360923121318429259338539

8*G
(x)2F01E5E15CCA351DAFF3843FB70F3C2F0A1BDD05E5AF888A67784EF3E10A2A01
(y)5C4DA8A741539949293D082A132D13B4C2E213D6BA5B7617B5DA2CB76CBDE904
(x)21262057306151627953595685090280431278183829487175876377991189246716355947009
(y)41749993296225487051377864631615517161996906063147759678534462689479575333124

--I apologize for the lousy formatting on the large numbers. I couldn't figure out how to reduce the font size.--

Answer 1

Your computations are correct.

For each valid $x$ coordinate there are 2 valid $y$ coordinate, $+y$ and $-y$, as you correctly said.

But note that numbers in a field $\mathbb{F}_p$ are usually written $\mod p$, which means they lie in the interval $[0,p-1]$, so without the "-" sign.

So, when you write the $y$ coordinate of your $-10G$, which you call $10*G (Inverse)$ but obviously $10*(-G)=-10*G$ so it's equivalent, you write it as: -62070622898698443831883535403436258712770888294397026493185421712108624767191 but this can be written as a field element as
p - 62070622898698443831883535403436258712770888294397026493185421712108624767191 with $p$ being the prime defining the field. This results into 53721466338617751591687449605251649140499096371243537546272162295800209904472 .

Plugging the coordinates into the curve's equation is the correct way to verify that points are on the curve, and yours are on the curve.

For example: your $2G-10G$ and $8G$ share the same $x$ coordinate, in fact, $2G-10G=-8G$ so they are opposite points. You can get the value of the $y$ if you subtract one's value from $p$. E.g.

p - 41749993296225487051377864631615517161996906063147759678534462689479575333124 
= 74042095941090708372193120377072390691273078602492804360923121318429259338539

Answer 2

According to this EllipticCurve code repo, you can validate a point is on the curve with the following function:

    public bool IsPointOnCurve(EllipticCurvePoint point)
    {
        if (point == EllipticCurvePoint.InfinityPoint)
        {
            return true;
        }

        BigInteger y = point.Y;
        BigInteger x = point.X;
        BigInteger a = this.A;
        BigInteger b = this.B;
        BigInteger p = this.P;

        var rem = (y * y - x * x * x - a * x - b) % p;

        return rem == 0;
    }

Plugging in your point coordinates, it looks like it's on the curve alright:

+       y   {41749993296225487051377864631615517161996906063147759678534462689479575333124} System.Numerics.BigInteger
+       x   {21262057306151627953595685090280431278183829487175876377991189246716355947009} System.Numerics.BigInteger
+       a   {0} System.Numerics.BigInteger
+       b   {7} System.Numerics.BigInteger
+       p   {115792089237316195423570985008687907853269984665640564039457584007908834671663}    System.Numerics.BigInteger
+       rem {0} System.Numerics.BigInteger